Find the positive value of $\lambda$ if the distance between the points $(7, 1, -3)$ and $(4, 5, \lambda )$ is $13$ units.

Let $P (7, 1, -3)$ and $Q (4, 5, A)$ be the given points.

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}PQ=\sqrt{9+16+(\lambda +3{)}^2}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{13}^2=25+(\lambda +3{)}^2\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\lambda }^2+6\lambda -135=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\lambda =-15,9\end{array}$